The stability conjecture for infinite set-homogeneous hypergraphs

Let $k\geq 3$, and let $H$ be an infinite set-homogeneous $k$-hypergraph whose theory is stable, meaning that it does not have the order property: there are no formula $\phi(\bar{x},\bar{y})$, model $M$, and sequences $(\bar{a}_i)_{i\in\omega}$ and $(\bar{b}_j)_{j\in\omega}$ such that $M\models\phi(\bar{a}_i,\bar{b}_j)$ if and only if $i<j$. A $k$-hypergraph is complete if every $k$-element subset is a hyperedge, and its complete complement is obtained by replacing the hyperedge relation with its complement.

The stability conjecture. Any infinite set-homogeneous $k$-hypergraph with stable theory is complete or has complete complement.

This predicts that stability forces an infinite set-homogeneous $k$-hypergraph to be trivial up to complementation. The supplied text poses the statement without indicating whether it is known or resolved.